Problem: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{n^3 - 10n^2 + 16n}{7n^2 - 49n - 56}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {n(n^2 - 10n + 16)} {7(n^2 - 7n - 8)} $ $ p = \dfrac{n}{7} \cdot \dfrac{n^2 - 10n + 16}{n^2 - 7n - 8} $ Next factor the numerator and denominator. $ p = \dfrac{n}{7} \cdot \dfrac{(n - 8)(n - 2)}{(n - 8)(n + 1)}$ Assuming $n \neq 8$ , we can cancel the $n - 8$ $ p = \dfrac{n}{7} \cdot \dfrac{n - 2}{n + 1}$ Therefore: $ p = \dfrac{ n(n - 2)}{ 7(n + 1)}$, $n \neq 8$